BME 315 Biomechanics, U. Wisconsin Adapted by R. Lakes from D. Thelen and C. Decker, 09, adapted from Lakes 06 Experimental Details I. Laboratory equipment The load frame that we will use to perform our testing is the MTS Sintech Universal Testing Machine set up in a 3-point bending configuration (Fig. 1). We will use the machine to provide a controlled, constant rate of deflection at the midpoint between the two supports. A load cell built into the crosshead will be used to measure the force applied. The system is interfaced with the laboratory data acquisition program TestWorks 4 (Fig. 2), providing for simultaneous measurement of deflection and load. Figure 1: Sintech 10/GL test frame configured for a 3-point bending test. Figure 2: TestWorks 4 interface.
II. Theoretical background 3-point bending is a common beam loading situation, with fixed supports under a beam and a point force in the middle. A free body diagram and the shear and moment diagrams for this loading demonstrate that the internal moment reaches a peak value at mid-span (Fig. 3). Figure 3: Shear force and bending moment (BM) diagrams for 3-point bending with load centered at midpoint [2]. Internal bending moments, M, induce axial stress, σ, in the beam that can be computed using the equation: where y is the distance from the neutral axis of the beam, and I is the area moment of inertia of the cross-section about the neutral axis. Unlike many engineered beams, bones do not have simple, constant cross-sections as assumed in simple beam theory. Further, the central cavity of this bone contains some spongy bone and bone marrow which would contribute somewhat to the strength of the bone. However for this lab, we will only consider the cortical bone and further will assume the beam cross-section can be approximated as concentric ellipses (Fig. 4). (1) Figure 4: Ellipse geometry for moment of inertia calculation. The area moment of inertia of a concentric elliptical cross-section (Fig. 4) is given by: (2)
From Eq. 1, it follows that the maximum normal stress is found at the maximum distance from the neutral axis. We will assume that the neutral axis passes through the midpoint of our specimen, and then set y = b o for calculating the maximum normal (axial) stress. The elastic equation relating the beam modulus of elasticity E, internal bending moment M, and deflection v along the length of a beam of constant cross-section is given by: Applying the boundary conditions for a 3-point bending configuration results in an expression for the maximum deflection at the mid-span of the beam (x=l/2) where L is the length of the beam (Fig. 5): (3) (4) In this lab, we will be measuring the beam deflection v and associated load P, which allows us to use Eq. 4 to estimate Young s modulus for cortical bone. Figure 5: Beam bending tables show maximum deflection for standard bending configurations [3]. III. Procedure details Obtaining a test specimen You will be working in teams of two to prepare samples and collect mechanical testing data; however, for the final lab write-up you will be using the data from the whole class. The dry bone specimens have been prepared before class, by exposing the bones to an extended drying period in an oven. You will each need to clean and test one wet bone sample. 1. Obtain a bone specimen and check to make sure it is thawed. Be sure to wear gloves and safety glasses. 2. Before performing the 3-point bending test, the muscle and connective tissue must be removed from the specimen as much as is possible. Use the scalpel and a safe technique (cutting away from yourself) to remove the tissue from the bone (Fig. 6).
BME 315 Biomechanics Figure 6: Utilize a safe technique (cutting away from yourself) when removing tissue from bone specimen. 3. When the tissue is fully removed from the bone, use the calipers to measure the maximum and minimum diameter at approximately the mid-span of the sample. With a three-point bending test, we should fail the specimen at the middle of the beam, and will need to know the moment of inertia at this location. The wall thickness will be measured after we break our specimen. Record your diameter measurements on the class data table, along with a name for your specimen. 4. Measurements also need to be made on the dry bone specimens. If your group has time, make these measurements and record this information on the class data sheet as well. 3-point bending test procedure MTS testing machines will be set up to perform our 3-point bending tests. These machines are very strong and can be dangerous, as they are capable of applying a load of more than 10,000 pounds. For this reason, it is important that you wear safety glasses and pay attention when a test is running. Stand a safe distance away from the equipment and do not attempt to adjust or move a sample while testing. Keep hands, arms and elbows away from test area. 1. The span length will depend on the kind of bone to be tested. A span of 40 mm is appropriate for chicken leg bones. Set up the specimen as shown (Fig. 7) when your TA indicates that the equipment is ready for a new test. Orient the sample such that the maximum diameter (the most stable surface) is oriented as the base (along axis X as shown in Fig. 4). Make sure that paper towels are present under the specimen as the adipose on the surface of the bone and marrow within it will be somewhat messy. 2. One or two people from each lab will assist in the operation of the MTS equipment. a. If they are not already on, turn on the MTS testing machine, followed by the computer controlling unit next to it. Select the user Students to bring up the desktop we will use for our testing.
b. To conduct a test, open the program TestWorks 4 from the computer desktop if it is not already open. The login name on the computer is student and there is no password. c. A dropdown window will open from which you should select the file 3PointBendBME315 from the folder SMTLab UW MADISON. If this dropdown window does not automatically open, you can find it by selecting Open Open Method from the top menu bar. d. You will indicate to the computer that you will use a new sample each time. To run a new sample select the icon that looks like dog bones (which represent standard tensile specimen bars) in the upper left corner (Fig. 2). e. To move the crosshead, first select Motor Reset from the lower right corner (Fig. 2). f. We first want to zero the load cell. Make sure the load cell is not contacting the specimen or anything else that might create a load. Click on the Force display in the lower right. Right mouse click to bring up a command window, and select Zero Channel to zero the load reading. g. When the specimen is set up approximately symmetrically with the diaphysis as the portion being tested (Fig. 7), carefully lower the crosshead close to the bone with the computer controls (Fig. 2). You may operate with the clutch in high when the load foot is far from the specimen, but you should use a low clutch when the distance is less than 1 cm. It is very easy to accidentally break the specimen when the clutch is in high. The clutch control is found in the lower right corner of the computer control panel (Fig. 2). h. Continue to lower the crosshead with the clutch in low until the Force display on screen shows that a small load of 10 20 N has been applied. i. Click on the Crosshead display in the lower left. Right mouse click to bring up a command window, and select Zero Channel to zero the displacement reading. j. You can now initiate the test. The computer will prompt you to enter the geometry of the specimen, however, it is not necessary to do this as the program is configured to make calculations for a standard compression or tension test, and not bending. We will be making our calculations independently after the test is complete. Make sure to enter the specimen name however. k. When the test begins, the force-deflection curve will display on the computer. Continue the test through failure (fracture) of the bone. Click Ok to return the crosshead to its original location. NOTE: The testing equipment may stop if it experiences a jump in the load. Testing equipment is commonly set up like this for safety purposes. If the specimen rotates or experiences a partial failure, the test equipment may stop. We partially load the specimen before beginning the test to try to prevent this from happening. If this happens, consult with your TA concerning what to do next. l. When the specimen has failed, save the data. To save the data select File>Export preview>specimen and save the data to the filename that the group selected. m. Use the computer controls to raise the crosshead. After the crosshead has been raised, the specimen can be removed. n. To perform the testing on a new sample, begin with Step 2.d and repeat the procedure. 3. Measure the wall thickness of the sample (Fig. 8) and record this information in the class data sheet, along with other comments concerning the failure. Comments should include if the test slipped, or if the sample did not fail at mid-span. If the failure was not at mid-span, further describe what occurred.
BME 315 Biomechanics 4. Pass the specimen around so that everyone in the class can take notes on the failure that you observe. Take photographs to document the failure as well. Figure 7: 3-point bending test of chicken leg bone, where d = crosshead deflection. Figure 8: Use calipers to measure wall thickness of cortical bone after specimen has experienced failure. IV. Analysis 1. Use Eq. 1 to calculate the maximum normal stress in the specimen. Plot the maximum normal stress versus the deflection for all of the specimen from your lab section. Include all specimens on the same figure.
2. Evaluate the normal stress versus deflection curves for each of the specimen and determine the yield strength (end of linear region of the stress/deflection curve) and the ultimate strength (maximum stress). Make a table (to be included in the Appendix of your report) showing the yield and ultimate strength for each of the specimens. 3. Calculate the average and standard deviation of yield strength and ultimate strength. If there are data that you feel are outliers (i.e. slippage within the test fixture, failure at a location other than midspan, etc.) do not include them in your calculation and justify why you chose to do so. 4. Rearrange Eq. 4 to solve for the modulus of elasticity E (Young s modulus). Determine the modulus of elasticity E (Young s modulus). Plot the modulus of elasticity versus deflection from the linear region of the stress versus deflection curve. Include all the specimens on one figure. Be sure to only use the portion of the data in which linearly elastic behavior occurs.calculate the average modulus of elasticity for each specimen (your plots will show a range) and include these values on your table. 5. Calculate the average and standard deviation of the modulus of elasticity for all of the specimen. Again, if there is an outlier, do not include this in your calculations and justify why you chose to do so. V. Questions 1. How do the values of Young s modulus, yield strength, and ultimate strength that you measured compare to values given in your book or other published data? Refer to Lakes [4], Fung [5], Shahnazari [6], and other references you may find. Comment on possible reasons for differences that exist. 2. Would you classify the wet bone as brittle or ductile? How about the dry bone? Why is there a difference in material properties between wet and dry bone? Why might one want to know the properties of both wet and dry bone? 3. Discuss your observations from the tests and the observed failures. Was the failure a result of tensile or compressive stresses? Why? 4. What are the main sources of error in our measurements? Consider some of the assumptions we made (cross-section geometry, neutral axis, contribution of trabecular bone, slenderness of specimen, etc.). How might this affect the calculations we have made? 5. How does the modulus, yield strength, and ultimate strength compare to other materials such as steel and titanium? What are some practical implications of this (i.e. if we are designing joint replacements or fixturing devices)? VI. References 1. Institute of Applied Mechanics at the University of Paderborn, Germany. Available at http://www.dhondt.de/examples.htm, accessed on 3/26/10. 2. Beardmore, R., Shear Force and Bending Moment Diagrams, RoyMech 2009. Available at http://www.roymech.co.uk/useful_tables/beams/shear_bending.html, accessed 6/1/10. 3. Hibbeler, R. C. Engineering Mechanics: Statics. Prentice-Hall, Inc., 1995. 4. Lakes, R., Anisotropy of Bone, University of Wisconsin Madison, Biomechanics BME 315. Available at http://silver.neep.wisc.edu/~lakes/bme315fr.html, accessed 5/17/10. 5. Fung, Y.C. (1993). Biomechanics: Mechanical Properties of Living Tissues. New York: Springer- Verlag New York, Inc. 6. Shahnazari, M., et al. Strontium Administration in Young Chickens Improves Bone Volume and Architecture but Does not Enhance Bone Structural and Material Strength, Calcif Tissue Int 80 (2007): 160 166.